Crossed Product C*-algebra

Idea

Roughly speaking, the crossed product $C^\ast$-algebra of a $C^\ast$-dynamical system $(A,G,\alpha)$ is the universal (not in the sense of ‘universal enveloping’) $C^\ast$-algebra $C$ that covariantly represent $(A,G,\alpha)$. All covariant representations of $(A,G,\alpha)$ factors through $C$.

The crossed product can also be seen as a twisted convolution that encodes the data of the action of $G$ on $A$, or as a $C^\ast$-algebra that carries the information of ‘quotient space’ $X/G$, where $X$ is the Isbell dual of $A$.

We shall present a definition via universal property, and then give an explicit definition.

Definition

Let $(A,G,\alpha)$ be a $C^\ast$-dynamical system. That is, a locally compact group $G$ and a homomorphism $\alpha:G\to \operatorname{Aut}A$.

A covariant representation of $(A,G,\alpha)$ in a $C^\ast$-algebra $B$ is a pair $(\pi,U)$ with $\pi\in\text{Mor}(A,B)$ and $U\in\text{Rep}(G,B)$ (note that this means $U$ is a unitary representation $U:G\rightarrow M(B):t\mapsto U_t$ and $M(B)$ is the multiplier), such that for any $a\in A$ and $t\in G$, $$ \pi\circ\alpha_t(a) = U_t \pi(a) U^\ast_t $$

A crossed product of $A$ by the action $\alpha$ of $G$ is a $C^\ast$-algebra $C$, with a covariant representation $(\pi,U)$ of $(A,G,\alpha)$ in $C$ s.t. for any $C^\ast$-algebra $B$ and any covariant representation $(\rho,V)$ of $(A,G,\alpha)$ in $B$ there is a unique $\Phi\in\text{Mor}(C,B)$ s.t. $\rho = \Phi\circ \pi$ and $V_t = \Phi(U_t)$ for all $t\in G$.

Properties

Uniqueness

From the definition it is clear that a crossed product defined from a $C^\ast$-dynamical system is unique up to isomorphism preserving the covariant representation, hence it makes sense to speak of the crossed product $A\rtimes_\alpha G$.

Existence

The crossed product of $A\rtimes_\alpha G$ always exists for a $C^\ast$-dynamical system $(A,G,\alpha)$, see Piotr Soltan, $C^\ast$-Algebras, group actions and crossed products, lecture notes.

Theorems

Explicit definition

The crossed product $A\rtimes_\alpha G$ of $C^\ast$-dynamical system $(A,G,\alpha)$ is the completion of the twisted convolution algebra $C_c(G,A)$ (which is a normed $\ast$-algebra, and whose completion in the $L^2$ norm is a Banach $\ast$-algebra) with respect to the supremum norm for the integrated forms of all covariant representations (called universal norm): $$ ||a|| = \text{sup}_{\pi,U} { ||\pi\rtimes_\alpha U(a)||:(\pi,U)\text{ is a covariant representation of }(A,G,\alpha)}. $$ For details, see P. Williams. Crossed products of C∗-algebras. Number 134. Providence: American Mathematical Soc., 2007 and Soltan’s note.

Some Remarks

The reason to consider covariant representations of $(A,G,\alpha)$ is that these induce $\ast$-representations of the convolution algebra $C_c(G,A)$.

Reduced Crossed Products

If $\pi_0$ is a faithful and non-degenerate representation of $A$ on a Hilbert space $\mathcal{H}$, and $(A,G,\alpha)$ is a $C^\ast$-dynamical system, then we can construct a covariant representation $(\pi,U)$ of $(A,G,\alpha)$ in $\mathcal{K}(H)$ where $H=\mathcal{H}\otimes L^2(G)$. In this case $\pi\rtimes_\alpha U$ is injective. Let $\Lambda\in \text{Mor}(A\rtimes_\alpha G,\mathcal{K}(H)$ be the extension of $\pi\rtimes_\alpha U$ from the convolution $C_c(G,A)$ to the whole $A\rtimes_\alpha G$, the image of $A\rtimes_\alpha G$ under the map $\Lambda$, corresponding to any faithful representation $\pi_0$ of $A$ is called the reduced crossed product of $A$ by the action $\alpha$ of $G$.

Note that

  1. The reduced crossed product $A\rtimes^r_\alpha G$ does not depend on the faithful representation $\pi_0$ of $A$.
  2. $\Lambda$ is not necessarily injective. Some partial results: if $G$ is amenable, then $\Lambda$ is injective, hence full and reduced crossed products are isomorphic.

When $A=\mathbb{C}$, we obtain the image of $C^\ast(G)$ under the left regular representation in $L^2(G)$. This $C^\ast$-algebra is called the reduced group $C^\ast$-algebra of $G$ and is denoted by $C^\ast_r(G)$.